docs: def spectral_embedding(geom, n_components=8, eigen_solver='auto',
                         random_state=None, drop_first=True,
                         diffusion_maps = False, diffusion_time = 0, solver_kwds = None):
      """
      Project the sample on the first eigen vectors of the graph Laplacian.
  
      The adjacency matrix is used to compute a normalized graph Laplacian
      whose  principal eigenvectors (associated to the
      smallest eigen values) represent the embedding coordinates of the data.
  
      The ``adjacency`` variable is not strictly the adjacency matrix of a graph but more generally
      an affinity or similarity matrix between samples (for instance the
      heat kernel of a euclidean distance matrix or a k-NN matrix).
      The Laplacian must be symmetric so that the eigen vector decomposition works as expected.
      This is ensured by the default setting (for more details,
      see the documentation in geometry.py).
  
      The data and generic geometric parameters are passed via a Geometry object, which also
      computes the Laplacian. By default, the 'geometric' Laplacian (or "debiased", or "renormalized" with
      alpha=1) is used. This is the Laplacian construction defined in [Coifman and Lafon, 2006] (see also
      documentation in laplacian.py). Thus, with diffusion_maps=False, spectral embedding is a modification
      of the Laplacian Eigenmaps algorithm of [Belkin and Nyiogi, 2002], with diffusion_maps=False, geom.laplacian_method
      ='symmetricnormalized' it is exactly the Laplacian Eigenmaps, with diffusion_maps=True, diffusion_time>0 it
      is the Diffusion Maps algorithm of [Coifman and Lafon 2006]; diffusion_maps=True and diffusion_time=0 is the same
      as diffusion_maps=False and default geom.laplacian_method.
  
      Parameters
      ----------
      geom : a Geometry object from megaman.embedding.geometry
      n_components : integer, optional
          The dimension of the projection subspace.
      eigen_solver : {'auto', 'dense', 'arpack', 'lobpcg', or 'amg'}
          'auto' :
              algorithm will attempt to choose the best method for input data
          'dense' :
              use standard dense matrix operations for the eigenvalue decomposition.
              For this method, M must be an array or matrix type.  This method should be avoided for large problems.
          'arpack' :
              use arnoldi iteration in shift-invert mode. For this method,
              M may be a dense matrix, sparse matrix, or general linear operator.
              Warning: ARPACK can be unstable for some problems.  It is best to
              try several random seeds in order to check results.
          'lobpcg' :
              Locally Optimal Block Preconditioned Conjugate Gradient Method.
              A preconditioned eigensolver for large symmetric positive definite
              (SPD) generalized eigenproblems.
          'amg' :
              AMG requires pyamg to be installed. It can be faster on very large,
              sparse problems, but may also lead to instabilities.
      random_state : int seed, RandomState instance, or None (default)
          A pseudo random number generator used for the initialization of the
          lobpcg eigen vectors decomposition when eigen_solver == 'amg'.
          By default, arpack is used.
      drop_first : bool, optional, default=True
          Whether to drop the first eigenvector. For spectral embedding, this
          should be True as the first eigenvector should be constant vector for
          connected graph, but for spectral clustering, this should be kept as
          False to retain the first eigenvector.
      diffusion_map : boolean, optional. Whether to return the diffusion map
          version by re-scaling the embedding coordinate by the eigenvalues to the power
          diffusion_time.
      diffusion_time: if diffusion_map=True, the eigenvectors of the Laplacian are rescaled by
          (1-lambda)^diffusion_time, where lambda is the corresponding eigenvalue.
          diffusion_time has the role of scale parameter. One of the main ideas of diffusion framework is
          that running the diffusion forward in time (taking larger and larger
          powers of the Laplacian/transition matrix) reveals the geometric structure of X at larger and
          larger scales (the diffusion process).
          diffusion_time = 0 empirically provides a reasonable balance from a clustering
          perspective. Specifically, the notion of a cluster in the data set
          is quantified as a region in which the probability of escaping this
          region is low (within a certain time t).
          Credit to Satrajit Ghosh (http://satra.cogitatum.org/) for description
      solver_kwds : any additional keyword arguments to pass to the selected eigen_solver
  
      Returns
      -------
      embedding : array, shape=(n_samples, n_components)
          The reduced samples.
  
      Notes
      -----
      Spectral embedding is most useful when the graph has one connected
      component. If there graph has many components, the first few eigenvectors
      will simply uncover the connected components of the graph.
  
      References
      ----------
      * http://en.wikipedia.org/wiki/LOBPCG
  
      * Toward the Optimal Preconditioned Eigensolver: Locally Optimal
        Block Preconditioned Conjugate Gradient Method
        Andrew V. Knyazev
        http://dx.doi.org/10.1137%2FS1064827500366124
      """
      random_state = check_random_state(random_state)
  
      if geom.affinity_matrix is None:
          geom.compute_affinity_matrix()
      if not _graph_is_connected(geom.affinity_matrix):
          warnings.warn("Graph is not fully connected: "
                        "spectral embedding may not work as expected.")
  
      if geom.laplacian_matrix is None:
          laplacian = geom.compute_laplacian_matrix(copy=False,
                                                    return_lapsym=True)
      else:
          laplacian = geom.laplacian_matrix
  
      n_nodes = laplacian.shape[0]
      lapl_type = geom.laplacian_method
      eigen_solver, solver_kwds = check_eigen_solver(eigen_solver,solver_kwds,
                                                     size=laplacian.shape[0],
                                                     nvec=n_components + 1)
      re_normalize = False
      PD_solver = False
      if eigen_solver in ['amg', 'lobpcg']: # these methods require a symmetric positive definite matrix!
          epsilon = 2
          PD_solver = True
          if lapl_type not in ['symmetricnormalized', 'unnormalized']:
              re_normalize = True
              # If lobpcg (or amg with lobpcg) is chosen and
              # If the Laplacian is non-symmetric then we need to extract:
              # the w (weight) vector from geometry
              # and the symmetric Laplacian = S.
              # The actual Laplacian is L = W^{-1}S  (Where W is the diagonal matrix of w)
              # Which has the same spectrum as: L* = W^{-1/2}SW^{-1/2} which is symmetric
              # We calculate the eigen-decomposition of L*: [D, V]
              # then use W^{-1/2}V  to compute the eigenvectors of L
              # See (Handbook for Cluster Analysis Chapter 2 Proposition 1).
              # However, since we censor the affinity matrix A at a radius it is not guaranteed
              # to be positive definite. But since L = W^{-1}S has maximum eigenvalue 1 (stochastic matrix)
              # and L* has the same spectrum it also has largest e-value of 1.
              # therefore if we look at I - L* then this has smallest eigenvalue of 0 and so
              # must be positive semi-definite. It also has the same spectrum as L* but
              # lambda(I - L*) = 1 - lambda(L*).
              # Finally, since we want positive definite not semi-definite we use (1+epsilon)*I
              # instead of I to make the smallest eigenvalue epsilon.
              if geom.laplacian_weights is None: # a laplacian existed but it wasn't called with return_lapsym = True
                  geom.compute_laplacian_matrix(copy = False, return_lapsym = True)
              w = np.array(geom.laplacian_weights)
              symmetrized_laplacian = geom.laplacian_symmetric.copy()
              if sparse.isspmatrix(symmetrized_laplacian):
                  symmetrized_laplacian.data /= np.sqrt(w[symmetrized_laplacian.row])
                  symmetrized_laplacian.data /= np.sqrt(w[symmetrized_laplacian.col])
                  symmetrized_laplacian = (1+epsilon)*sparse.identity(n_nodes) - symmetrized_laplacian
              else:
                  symmetrized_laplacian /= np.sqrt(w)
                  symmetrized_laplacian /= np.sqrt(w[:,np.newaxis])
                  symmetrized_laplacian = (1+epsilon)*np.identity(n_nodes) - symmetrized_laplacian
          else: # using a symmetric laplacian but adjust to avoid positive definite errors
              symmetrized_laplacian = geom.laplacian_matrix.copy()
              if sparse.isspmatrix(symmetrized_laplacian):
                  symmetrized_laplacian = (1+epsilon)*sparse.identity(n_nodes) - symmetrized_laplacian
              else:
                  symmetrized_laplacian = (1+epsilon)*np.identity(n_nodes) - symmetrized_laplacian
  
      if PD_solver: # then eI - L was used, fix the eigenvalues
          lambdas, diffusion_map = eigen_decomposition(symmetrized_laplacian, n_components+1, eigen_solver=eigen_solver,
                                                       random_state=random_state, drop_first=drop_first, largest = False,
                                                       solver_kwds=solver_kwds)
          lambdas = -lambdas + epsilon
      else:
          lambdas, diffusion_map = eigen_decomposition(laplacian, n_components+1, eigen_solver=eigen_solver,
                                                       random_state=random_state, drop_first=drop_first, largest = True,
                                                       solver_kwds=solver_kwds)
      if re_normalize:
          diffusion_map /= np.sqrt(w[:, np.newaxis]) # put back on original Laplacian space
          diffusion_map /= np.linalg.norm(diffusion_map, axis = 0) # norm 1 vectors
      # sort the eigenvalues
      ind = np.argsort(lambdas); ind = ind[::-1]
      lambdas = lambdas[ind]; lambdas[0] = 0
      diffusion_map = diffusion_map[:, ind]
      eigenvalues = lambdas.copy()
      eigenvectors = diffusion_map.copy()
      if diffusion_maps:
          diffusion_map = compute_diffusion_maps(lapl_type, diffusion_map, lambdas, diffusion_time)
      if drop_first:
          embedding = diffusion_map[:, 1:(n_components+1)]
          eigenvectors = eigenvectors[:, 1:(n_components+1)]
          eigenvalues = eigenvalues[1:(n_components+1)]
      else:
          embedding = diffusion_map[:, :n_components]
          eigenvectors = eigenvectors[:, :(n_components)]
          eigenvalues = eigenvalues[:(n_components)]
      return embedding, eigenvalues, eigenvectors